Diagonally-reduced tensor-product polynomials: dimensions and boundary-diagonal matchingGradišek, Domen (Avtor) Grošelj, Jan (Avtor) bivariate polynomialstensor-product polynomialsdiagonal curvesboundary-diagonal matchingBézier-smart surfacesserendipity elementsWe study subspaces of tensor-product polynomials of bi-degree $(n, n)$ on the unit square whose restrictions to the domain diagonals have degree reduced by an integer $k \le n$. For maximal reduction $k = n$, these diagonally-reduced spaces coincide with the Bézier-Smart surface spaces and have dimension $n^2 + 2$. We derive exact dimension formulas and solve the boundary-diagonal matching problem: when can such a polynomial be replaced by a total-degree polynomial that agrees on the boundary and both diagonals? The answer is a sharp threshold on the degree parameters, with a complete characterisation of existence, uniqueness, and non-uniqueness. For the first non-trivial reduction parameters we provide explicit matching formulas and closed-form error estimates; when the solution is non-unique, the remaining freedom is resolved by $L^2$ optimality. At the endpoint $k = n$, where matching always fails, we show for $n = 3$ that the $L^2$-best approximation from the total-degree subspace is also the $L^\infty$-best approximation.20272026-07-01 10:45:01Neznano30698sl